Skew Killing spinors

We study the existence of a skew Killing spinor on 2- and 3-dimensional Riemannian spin manifolds. We establish the integrability conditions and prove that these spinor fields correspond to twistor spinors in the two dimensional case while, up to a conformal change of the metric, they correspond to parallel spinors in the three dimensional case.     

The length function of a twistor spinor with zero

We examine the Taylor expansion of the length function of a twistor spinor with zero on a Riemannian orbifold around its zero and study it on the Eguchi–Hanson orbifold. This expansion is written in some conformal normal coordinates (CNC) around the zero up to order 7. In the example of the Eguchi–Hanson orbifold, CNC are found explicitly. We use the expansion in computing the mass (a generalization of ADM–mass) of the asymptotically locally Euclidean coordinate system, which is constructed from a conformal normal coordinate system around the zero of a twistor spinor on a Riemannian spin orbifold admitting isolated singularities.     

Causal conformal vector fields,and singularities of twistor spinors

In this paper, we study the geometry around the singularity of a twistor spinor, on a Lorentz manifold (M, g) of dimension greater or equal to three, endowed with a spin structure. Using the dynamical properties of conformal vector fields, we prove that the geometry has to be conformally flat on some open subset of any neighbourhood of the singularity. As a consequence, any analytic Lorentz manifold, admitting a twistor spinor with at least one zero has to be conformally flat.      

Absence of Differential Correlations Between the Wave Equations for Upper-Lower One-Index Twistor Fields Borne by the Infeld-van derWaerden Spinor Formalisms for General Relativity

It is pointed out that the wave equations for any upper-lower one-index twistor fields which take place in the frameworks of the Infeldvan der Waerden γε-formalisms must be formally the same. The only reason for the occurrence of this result seems to be directly related to the fact that the spinor translation of the traditional conformal Killing equation yields twistor equations of the same form. It thus appears that the conventional torsionless devices for keeping track in the γ-formalism of valences of spinor differential configurations turn out not to be useful for sorting out the typical patterns of the equations at issue.     

Scalar Curvature Rigidity for Asymptotically Locally Hyperbolic Manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the four dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.     

Compact anti-self-dual orbifolds with torus actions

We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat K?hler orbifold. We use this observation to classify ALE scalar-flat K?hler 4-orbifolds whose isometry group contain a 2-torus.     

Generalized Killing spinors on spheres

We study generalized Killing spinors on round spheres \(\mathbb {S}^n\) . We show that on the standard sphere \(\mathbb {S}^8\) any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on \(\mathbb {S}^n\) whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.     

Geometric aspects of transversal Killing spinors on Riemannian flows

We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those flows carrying non-trivial solutions.      

The twistor equation in Lorentzian spin geometry

In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to dimension 7.Mathematics Subject Classification (2000): 53C15, 53C50in final form: 1 October 2003     

Conformal reference frames for Lorentzian manifolds

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, we define the celestial transformation of Lorentzian vectors, a kind of spinor correspondence. We express a 1-form generating the contact structure in the twistor space (when it is smooth) explicitly as a form taking line-bundle values. We prove a theorem on the projection of this 1-form to the fiberwise normal bundle of a reference frame; its corollary is an equation for the flow of time.     

Skew Killing spinors in four dimensions

Annals of Global Analysis and Geometry - This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $$\psi $$ is a...     

On a horizontal conformal Killing tensor of degreep in a sasakian space

Summary We deal with a horizontal conformal Killing tensor of degree p in a Sasakian space. After some preparations we prove that a horizontal conformal Killing tensor of odd degree is necessarily Killing. Moreover, we consider horizontal conformal Killing tensor of even degree. The form of the associated tensor is determined completely and a decomposition theorem is proved. Then we give the examples of a conformal Killing tensor of even degree and a special Killing tensor of odd degree with constant l. Entrata in Redazione il 17 luglio 1971.     

Conformal positive mass theorems for asymptotically flat manifolds with inner boundary

Inspired by Witten's insightful spinor proof of the positive mass theorem, in this paper, we use the spinor method to derive higher dimensional type conformal positive mass theorems on asymptotically flat spin manifolds with inner boundary, which states that under a condition about the plus (minus) relation between the scalar curvatures of the original and the conformal metrics in addition with some boundary condition, we will get the associated positivity of their ADM masses. The rigidity part of the plus part is used in the proof of black hole uniqueness theorems. They are related with quasi-local mass and the spectrum of Dirac operator.     

Canonical Lorentzian spin structure and twistor spinors on the Fefferman space of a contact Riemannian manifold

The Fefferman space of a contact Riemannian manifold carries a Lorentzian spin structure canonically. On the Lorentzian spin manifold, we investigate the Dirac operator and the twistor operator closely. In particular, we show that, if the contact Riemannian manifold is integrable, then there exist non-zero global solutions of the twistor equation.     

Killing spinors on K?hler manifolds

In the paper Kählerian Killing spinors are defined and their basic properties are investigated. Each Kähler manifold that admits a Kählerian Killing spinor is Einstein of odd complex dimension. Kählerian Killing spinors are a special kind of Kählerian twistor spinors. Real Kählerian Killing spinors appear for example, on closed Kähler manifolds with the smallest possible first eigenvalue of the Dirac operator. For the complex projective spaces P ^2l–1 and the complex hyperbolic spaces H ^2l–1 withl>1 the dimension of the space of Kählerian Killing spinors is equal to ( ). It is shown that in complex dimension 3 the complex hyperbolic space H ³ is the only simple connected complete spin Kähler manifold admitting an imaginary Kählerian Killing spinor.     

On conformal Killing symmetric tensor fields on Riemannian manifolds

A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds.     

On a conformal Killingp-form in a compact sasakian space

Summary Let M be a compact Sasakian space admitting a conformal Killing p-form u. Then, we show that the associated form ϑ of a conformal Killing form u is a special Killing form with constant 1. Moreover we prove the decomposition theorem of u and seek the condition for M to be a unit sphere. Entrata in Redazione il 29 agosto 1971. Delicated to ProfessorT. Adati on his 60th Birthday.     

A new construction of anti-self-dual four-manifolds

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D ₂ V = 0 on a five-dimensional manifold Y{\mathbf{Y}}. The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in {V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space Z{\mathcal{Z}} has dim|-\frac12K_Z| 3 2{ \dim\left|-\frac{1}{2}K_\mathcal{Z}\right|\geq 2}, projecting thus over \mathbb C\mathbb P₂{\mathbb C\mathbb P_2} with twistor lines mapping onto plane conics.